Abstract

If R is commutative ring then R can be embedded into a strongly $\pi$-regular ring if and only if there exists a set of prime ideals Y = $\{$P$\sb\alpha\}\alpha\in\Lambda$ and for each P$\sb\alpha$, a P$\sb\alpha$-primary ideal Q$\sb\alpha$ such that: (I) Y is closed in the patch topology on Spec R. (II) $\rm\cap\sb{\alpha\in\Lambda}Q\sb\alpha=\{0\}$. (III) for each a $\in$ R there is n(a) $\in$ N such that for n $\ge$ n(a), $\rm\{P\sb\alpha\mid P\sb\alpha\in Y$ and $\rm a\sp{n}\in Q\sb\alpha\}$ is patch open in Y.